Problem 44 - Algebra 2 and Trigonometry

Question

The dimensions of a rectangular solid are \(\sqrt{5}\) inches by \((2+\sqrt{3})\) inches by \((2-\sqrt{3})\) inches. Express the volume of the solid in simplest form.

Step-by-Step Solution

Answer

The volume of the solid is \(\sqrt{5}\).

Step 1: Write the Formula for Volume

The volume of a rectangular solid can be found using the formula: \[V = l \times w \times h\] where \(l\), \(w\), and \(h\) are the length, width, and height of the rectangular solid respectively.

Step 2: Substitute Dimensions

Substitute the given dimensions for length, width, and height into the volume formula: \[ V = \sqrt{5} \times (2 + \sqrt{3}) \times (2 - \sqrt{3}) \].

Step 3: Simplify the Product of Width and Height

Use the difference of squares formula \[(a+b)(a-b) = a^2 - b^2\] to simplify \[(2 + \sqrt{3})(2 - \sqrt{3})\]. This gives: \[ (2 + \sqrt{3})(2 - \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1 \].

Step 4: Calculate the Volume

Substitute the simplified expression from Step 3 into the volume expression: \[V = \sqrt{5} \times 1 = \sqrt{5}\].Thus, the volume of the rectangular solid is \(\sqrt{5}\).

Key Concepts

Rectangular Solid Dimensions

When we talk about the dimensions of a rectangular solid, it's important to remember that these refer to its length, width, and height. These three measurements define the shape and size of the solid. In the exercise given, the dimensions are slightly more complex because they include square roots and additions.

• Length: \(\sqrt{5}\) inches
• Width: \((2 + \sqrt{3})\) inches
• Height: \((2 - \sqrt{3})\) inches

These measurements may seem complicated, but they are really just three distinct lengths that describe our solid. Understanding how to manipulate these dimensions mathematically is crucial for finding the volume, which we will explore next.

Volume Formula

To find the volume of any rectangular solid, you use the volume formula: \[ V = l \times w \times h \]Where:

• \(V\) is the volume
• \(l\) is the length
• \(w\) is the width
• \(h\) is the height

The formula multiplies the three dimensions together to calculate the total volume of space occupied by the solid. It's important to substitute the right values into the formula, as done in the exercise. For this problem, substituting the values gives us:\[ V = \sqrt{5} \times (2 + \sqrt{3}) \times (2 - \sqrt{3}) \]This sets us up to begin simplifying the expression.

Difference of Squares

The concept of difference of squares is a valuable algebraic tool that simplifies many kinds of expressions. It states that:\[(a + b)(a - b) = a^2 - b^2\]This means when you multiply a sum and a difference of the same two terms, you can express the product as the difference between their squares. In the exercise:

• \(a = 2\)
• \(b = \sqrt{3}\)

Applying this formula to \((2 + \sqrt{3})(2 - \sqrt{3})\), we calculate:\[ (2)^2 - (\sqrt{3})^2 = 4 - 3 = 1 \]This simplifies our expression significantly, reducing a complex binomial multiplication to a single number. It's a critical step to make calculations cleaner and easier.

Simplifying Radicals

Simplifying radicals involves breaking down expressions that contain square roots into their simplest form. When expressing a number like \(\sqrt{5}\), it already represents its simplest radical form because 5 is a prime number. Unlike integers or fractions, you can't add or remove factors under the square root sign unless they form perfect squares.
In our exercise, once the width and height product was simplified to 1, we were left with:\[ V = \sqrt{5} \times 1 = \sqrt{5} \]Since \(\sqrt{5}\) cannot be simplified further, it is the final and simplest form of the volume for the rectangular solid. Mastering the skill to check radicals for simplification ensures precision in final solutions.